Hydrogeology and hydrology are the fields you're looking at here. It's not as straightforward as you might imagine.

So yes there are models, but they are of varying precision in different settings, and dependent on the quality of your subsurface knowledge.

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Modflow comes to mind as a modelling tool. Note that a lot of things influence this process. Initial state, to what extent is the soil already saturated, where is the groundwater table? What is the hydraulic conductivity of the subsurface? Vegetation, surface temperature, wind, humidity, incoming radiation all influence evapotranspiration losses too.

Yes, there are several models and methods available to calculate how groundwater distributes spatially. These include the analytical element method (AEM), finite element method (FEM), groundwater flow model (GWM), and the Darcy-Weisbach equation. Each of these methods have their own advantages and disadvantages and can be used to accurately model groundwater flow in different scenarios.

You could use local groundwater tables and soil types to predict where it might spread. You could also build a series of test wells and add some sort of marker to the water, wouldn't be that accurate, but high confidence.

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There is no one model. And in many aquifers no one much knows where or how water moves. So long as it's there to be tapped it's ignored.

The aquifers that are better modeled are the ones that have problems affecting the people using them. Regionally here that's the Odessa - overtapped for irrigation. Several underlying the Hanford site - contaminated with radioactive and chemical waste threatening a city's wells. And in the 1000 springs aquifer in Idaho - where aquaculture, agriculture and river flow needs for salmon collide.

There is a pretty good idea of where a liter of water dumped onto the ground is going to go, how long it's going to take to get there in those places.

But each one has entirely unique hydrology and geology. Their models don't interchange.